We construct higher-order Darboux transformations for Schrödinger equations with quadratically energy-dependent potentials. Similar to the case of conventional Darboux (also known as supersymmetric) transformations, both the solutions and the potentials of transformed equations are expressed through Wronskians. We discuss properties of our Darboux transformations and provide an application.
REFERENCES
1.
G.
Darboux
, “Sur une proposition relative aux équations linéaires
,” C. R. Acad. Sci.
94
, 1456
–1459
(1882
).2.
C.
Gu
, A.
Hu
, and Z.
Zhou
, Darboux Transformations in Integrable Systems
(Springer Science and Business Media
, Dordrecht
, 2005
).3.
V. G.
Bagrov
and B. F.
Samsonov
, “Darboux transformation, factorization, and supersymmetry in one-dimensional quantum mechanics
,” Theor. Math. Phys.
104
, 1051
(1995
).4.
F.
Cooper
, A.
Khare
, and U.
Sukhatme
, “Supersymmetry and quantum mechanics
,” Phys. Rep.
251
, 267
–388
(1995
).5.
D. J.
Fernandez
, “Supersymmetric quantum mechanics
,” AIP Conf. Proc.
1287
, 3
–36
(2010
).6.
D. J.
Fernandez
, “Trends in supersymmetric quantum mechanics
,” in , edited by Ş.
Kuru
, J.
Negro
and L.
Nieto
(Springer
, Cham
, 2019
).7.
M. A.
Garcia-Ferrero
, D.
Gomez-Ullate
, and R.
Milson
, “A Bochner type classification theorem for exceptional orthogonal polynomials
,” J. Math. Anal. Appl.
472
, 584
(2019
); arXiv:1603.04358.8.
V. V.
Kravchenko
and S.
Torba
, “Transmutations for Darboux transformed operators with applications
,” J. Phys. A: Math. Theor.
45
, 075201
(2012
).9.
A.
Contreras-Astorga
and A.
Schulze-Halberg
, “On integral and differential representations of Jordan chains and the confluent supersymmetry algorithm
,” J. Phys. A: Math. Theor.
48
, 315202
(2015
).10.
A.
Schulze-Halberg
, “Arbitrary-order Jordan chains associated with quantum-mechanical Hamiltonians: Representations and integral formulas
,” J. Math. Phys.
57
, 023521
(2016
).11.
J.
Lin
, Y.-S.
Li
, and X.-M.
Qian
, “The Darboux transformation of the Schrödinger equation with an energy-dependent potential
,” Phys. Lett. A
362
, 212
(2007
).12.
J.
Formanek
, R.
Lombard
, and J.
Mares
, “Wave equations with energy-dependent potentials
,” Czechoslov. J. Phys.
54
, 289
–315
(2004
).13.
A.
Schulze-Halberg
, “Higher-order Darboux transformations for the massless Dirac equation at zero energy
,” J. Math. Phys.
60
, 073505
(2019
).14.
A.
Schulze-Halberg
, “Darboux transformations for the massless Dirac equation with matrix potential: Radially symmetric zero-energy states
,” Eur. Phys. J. Plus
134
, 295
(2019
).15.
A.
Schulze-Halberg
and M.
Ojel
, “Darboux transformations for the massless Dirac equation with matrix potential: Construction of zero-energy states
,” Eur. Phys. J. Plus
134
, 49
(2019
).16.
A.
Schulze-Halberg
, “Darboux transformations for energy-dependent potentials and the Klein-Gordon equation
,” Math. Phys.: Anal. Geom.
16
, 179
(2013
).17.
N. V.
Ustinov
and S. B.
Leble
, “Korteweg-de Vries–modified Korteweg-de Vries systems and Darboux transforms in 1+1 and 2+1 dimensions
,” J. Math. Phys.
34
, 1421
(1993
).18.
A.
Schulze-Halberg
and M.
Paskash
, “Wronskian representation of second-order Darboux transformations for Schrödinger equations with quadratically energy-dependent potentials
,” Phys. Scr.
95
, 015001
(2020
).19.
P. J.
Browne
and R.
Nillsen
, “The two-sided factorization of ordinary differential operators
,” Can. J. Math.
32
, 1045
(1980
).20.
K.
Wolsson
, “A condition equivalent to linear dependence for functions with vanishing Wronskian
,” Linear Algebra Appl.
116
, 1
(1989
).21.
M.
Abramowitz
and I.
Stegun
, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables
(Dover Publications
, New York
, 1964
).© 2020 Author(s).
2020
Author(s)
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